A critical comparison of nonlocal and gradient-enhanced softening continua

Continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients. These gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators. Adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations. The effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting. A nonlocal damage model and two different gradient damage models are considered. In one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly, while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation. The latter, implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal. This is not true for the explicit formulation, in which the nonlocality is limited to an infinitesimal volume. This fundamental difference between the formulations results in quite different behaviour in wave propagation, localisation and at crack tips. This is shown for the propagation of waves in the models, their localisation properties and the behaviour at a crack tip. The responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations, while the explicit gradient formulation shows an entirely different and sometimes nonphysical response.